4 Answers
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If you are following someone on Twitter and they tweet something you want to share with the people following you, you retweet. This reposts the tweet onto your timeline and makes it possible for your followers (who may not be following the original tweeter) to see the tweet.

If you are following someone on Twitter, they tweet something and you reply to that tweet, the people following you can see your reply, but they don't know what the original tweet said.

If you choose to reply with a quote, then the original tweet will appear in your timeline as well. (Not all Twitter apps will allow you to reply with a quote.)

If we assume twitter as a mailbox, then a retweet would be a forward mail. A forward mail just some nice photos, jokes or some other interesting things which you got from someone and forwarding it to your contacts.

In the same scenario, if you reply back to that forward mail or some other mail which has been addressed to you it can be compared to a reply on twitter.

Now, those are what reply or retweet mean. Visually, old style retweet (refer Al's answer for old and new retweet styles) and reply are almost the same except the RT. Many people prefer the old style retweet as it allows you to comment/reply while you RT it.

Typically, a "retweet" is what you call it when you're merely quoting someone else's tweet with the purpose of sharing what they said with your friends list. A "reply" is not a quote, but rather a response to someone else's tweet.